"Schools of thought" seems generous. Most of the opinion on the subject (aside from reasonable criticism of cack-handed implementation) is shallow and visceral.
That said, most of the evidence that at least has the dressing of empiricism favours Remain. There are two things that reliably help an economy, immigration and lower barriers to trade; Brexit damages them both, in favour of up-sides based largely on ideas relating to jam, so economists have fairly uniformly come down against it.
Many on the leave side would (and did) say that while the EU/EEA has succeeded in lowering some barriers to trade between EEA states, it has at the same time (1) raised new barriers to internal trade within some EEA states and (2) maintained or raised new barriers to trade between EEA states and states outside the EEA.
This is particularly stark when new member states have acceded and as a result new tariff barriers put in place, as was the case when the UK joined the then EEC (this caused significant damage to the UK's relationship with New Zealand, for example).
But it's also true in the way that the EU/EEA views product regulations and standards, which (many would contend) have a protectionist effect. One example commonly cited in this regard is REACH.
Many leave campaigners expressed a desire to deregulate in these areas in order to lower barriers to trade in ways that were not possible within the EU/EEA, as well as to lower tariffs unilaterally post-Brexit. I don't think it's fair therefore to say that Brexit damages "lower barriers to trade" - it really depends on what kind of Brexit there is whether that's true or not.
Again, there are shades of opinion on this. There are many leavers who are all for lowered barriers to trade and recognise the stranglehold the European Economic Area has on standards. I'd be quite happy in the EEA, for instance. Your argument is a little reductive and black/white.
Standards are required for lowered barriers to trade - lack of standards are what cause people to raise barriers in the first place.
For example, take labour standards. Without common agreement on labour standards, it's relatively easy for one country to undercut another by permitting mistreatment of their labour force. The response will either be to lower labour standards as well (meaning the result is no net gain in terms of market share, but a whole lot of labour in a worse position), or put in place tariffs or other trading restrictions to protect the local labour force.
Thus common agreement on some level of labour standards is what permits lowered barriers to trade. It prevents beggar-thy-neighbour situations that make the majority of people worse off.
It's similar for environmental standards, food standards, product quality, intellectual property including trademarks and PDOs and PGIs that simulate them for traditional producers.
The EU single market wouldn't be possible without lots of consensus on standards. Remove the enforced harmonization of standards, and the market doesn't work.
(The EU's growth in centralized power via the single market resembles in many ways the growth in US federal power due to the Commerce Clause. It definitely has dangers when market regulation runs ahead of democratic consent. I personally think there is a European polis which understands our position in the world, but I also don't think that polis is a majority. Issues that drive a wedge between the polis and the rest - most especially antipathy towards immigration - is the biggest strategic weakness the EU has, geopolitically.)
Chlorinated chicken and the reasons some markets don't permit it is rather misunderstood - it's a potential safety issue for the workers who produce it, not the people who eat it.
